The F-Distributions


The F-distributions can be used to analyze the ratio of variances of two independent normally distributed populations. In particular, we can construct confidence intervals and perform hypothesis tests on such ratios with this distribution. ANOVA tests also make use of F-distributions.

An F-distribution is based on two parameters known as the degrees of freedom of the numerator m and of the denominator n, where m and n are integers greater than or equal to 1. Such a random variable is denoted by X ~ F(m,n). The probability density function (pdf) is given by

f(x) = D x^(m / 2 - 1) / (1+ mx / n) ^ ( (m + n) / 2 ) ,

for x >= 0, where D is a constant that depends on m and n.

For m >= 3, the graph of the F(n,m) pdf is a non-symmetric, skewed, "Bell-Shaped" curve defined for x >= 0. These graphs attain a maximum value at x = (mn - 2n) / (mn + 2m).

Using the FDIST Program

The FDIST program can be used to compute probabilities such as P(F(n,m) <=k), P(F(n,m) >= k), or P(j <= F(n,m) <= k). To execute the program, first enter 1, 2, or 3 to designate the type of probability you wish to compute. Next enter the degrees of freedom of the numerator, the degrees of freedom of the denominator, followed by the lower and upper bounds j and k, with 0 <= j <= k, (or enter just the single bound k for a tail probability). The program next asks if you want to see a shaded graph of the pdf. If so, enter 1. If not, then enter 0. After the graph, we receive a display of the desired probability.

Example. Let X ~ F(20, 25). Find the probability that X lies between 0.5 and 1.

Solution. Call up the FDIST program, enter 3 for MIDDLE PROB, then enter 20 for DEGREES OF NUM., 25 for DEGREES OF DEN., .5 for LOWER BOUND, and 1 for UPPER BOUND. We find that P(0.5 <= F(20, 25) <= 1) is about 0.4475.

Other Features:

1. If you entered 1 to see a graph, then the partially shaded graph of the pdf initially appears, but then is replaced by the display of the computed probability. To see the graph again, press GRAPH. Then to see the probability value again, press CLEAR. On the TI-86, press CLEAR twice, or just press EXIT to remove the graph. To exit the graph on the TI-89, press HOME. Then press F5 to see the program output again.

2. If you initially enter either 1 or 2 to compute a tail probability, then both the left-tail and right-tail probabilities will be displayed in either case. However only the desired region will be shaded in the graph.


Compute P(F(21,16) <= 1.2) and P(F(21,16) > 1.2).


After calling up the FDIST program, enter 1 for LEFT PROB, then enter the degrees of freedom followed by 1.2 for BOUND. We find that P(F(21,16) <= 1.2) = 0.6406274 and P(F(21,16) > 1.2) = 0.3593726.

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